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Mirrors > Home > MPE Home > Th. List > xmulpnf2 | Structured version Visualization version GIF version |
Description: Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmulpnf2 | ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 11275 | . . . 4 ⊢ +∞ ∈ ℝ* | |
2 | xmulcom 13252 | . . . 4 ⊢ ((+∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (+∞ ·e 𝐴) = (𝐴 ·e +∞)) | |
3 | 1, 2 | mpan 687 | . . 3 ⊢ (𝐴 ∈ ℝ* → (+∞ ·e 𝐴) = (𝐴 ·e +∞)) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = (𝐴 ·e +∞)) |
5 | xmulpnf1 13260 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (𝐴 ·e +∞) = +∞) | |
6 | 4, 5 | eqtrd 2771 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 0 < 𝐴) → (+∞ ·e 𝐴) = +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 0cc0 11116 +∞cpnf 11252 ℝ*cxr 11254 < clt 11255 ·e cxmu 13098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulcom 11180 ax-i2m1 11184 ax-rnegex 11187 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-xmul 13101 |
This theorem is referenced by: xmulrid 13265 xmulgt0 13269 xmulasslem3 13272 xlemul1a 13274 xadddi2 13283 nmoix 24566 nn0xmulclb 32417 hashxpe 32452 esumpinfsum 33539 |
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